CN106894328B - A kind of processing method of Π shape bondbeam Shear Lag - Google Patents
A kind of processing method of Π shape bondbeam Shear Lag Download PDFInfo
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- E—FIXED CONSTRUCTIONS
- E01—CONSTRUCTION OF ROADS, RAILWAYS, OR BRIDGES
- E01D—CONSTRUCTION OF BRIDGES, ELEVATED ROADWAYS OR VIADUCTS; ASSEMBLY OF BRIDGES
- E01D11/00—Suspension or cable-stayed bridges
- E01D11/04—Cable-stayed bridges
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- E—FIXED CONSTRUCTIONS
- E01—CONSTRUCTION OF ROADS, RAILWAYS, OR BRIDGES
- E01D—CONSTRUCTION OF BRIDGES, ELEVATED ROADWAYS OR VIADUCTS; ASSEMBLY OF BRIDGES
- E01D19/00—Structural or constructional details of bridges
- E01D19/12—Grating or flooring for bridges; Fastening railway sleepers or tracks to bridges
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/10—Geometric CAD
- G06F30/13—Architectural design, e.g. computer-aided architectural design [CAAD] related to design of buildings, bridges, landscapes, production plants or roads
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- E—FIXED CONSTRUCTIONS
- E01—CONSTRUCTION OF ROADS, RAILWAYS, OR BRIDGES
- E01D—CONSTRUCTION OF BRIDGES, ELEVATED ROADWAYS OR VIADUCTS; ASSEMBLY OF BRIDGES
- E01D2101/00—Material constitution of bridges
- E01D2101/20—Concrete, stone or stone-like material
- E01D2101/24—Concrete
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- E—FIXED CONSTRUCTIONS
- E01—CONSTRUCTION OF ROADS, RAILWAYS, OR BRIDGES
- E01D—CONSTRUCTION OF BRIDGES, ELEVATED ROADWAYS OR VIADUCTS; ASSEMBLY OF BRIDGES
- E01D2101/00—Material constitution of bridges
- E01D2101/30—Metal
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
- G06F2119/06—Power analysis or power optimisation
Abstract
The invention discloses a kind of processing methods of Π shape bondbeam Shear Lag, this method is directed to Π shape composite beam section construction features, reasonable assumption Shear Lag warp displacement mode, and the equilibrium equation of variational form is established based on minimum potential energy principal, by the positional displacement interpolation function for introducing every two Shear Lag freedom degrees of node, the variation equilibrium equation of continuous shape is converted to the finite segment method method of discrete shape, to acquire the element stiffness matrix for considering that Shear Lag influences.Thus related application is write, fining analysis is carried out to Π shape bondbeam Construction of Cable-Stayed Bridges, this method is capable of handling in cable-stayed bridge the mutation of Shear Lag displacement boundary conditions caused by concentrated bending moment, the stress in work progress middle girder section is really reflected, the accuracy of Π shape bondbeam Construction of Cable-Stayed Bridges is improved.Mechanical concept of the invention is clear, it is simple to calculate, and has a extensive future, is the useful supplement to Shear Lag processing method in current specifications.
Description
Technical field
The invention belongs to bridge construction control technical fields, and in particular to one kind Π shape in CONSTRUCTION OF CABLE-STAYED BRIDGE control process
The processing method of bondbeam Shear Lag Effect.
Background technique
With the rapid development of the national economy, the requirement to cable-stayed bridge span ability is higher and higher, therefore in the design more
More Π shape bondbeams using steel truss girder, prestressed concrete П ellbeam or " double I-steel+concrete slabs ".Π shape bondbeam
Can give full play to two kinds of materials of steel and concrete respectively advantage and convenient for construction, have extensive development prospect.However, Π shape knot
" the ratio of width to height " for closing beam is relatively very big, and top plate width is cut much larger than top plate width between box beam median ventral plate between especially two webs
Power hysteresis effect is the important mechanical characteristics of this opening wide cut thin walled beam.It is used in the higher cable-stayed bridge of Construction control difficulty
When Π shape bondbeam, accurately influence of the analysis Shear Lag to girder bending stiffness and section stress is extremely important, concerning knot
Structure safety and control precision.
For the research of thin walled beam Shear Lag Effect, there is significant progress.《It highway reinforced concrete and answers in advance
Power concrete bridges and culverts design specification》(JTG D62-2004, P16~18) 4.2.2 and 4.2.3 articles respectively cut T shape and box-shaped
The Studies On Effective Width of face beam is provided, the influence of Shear Lag is considered with this, however this method is primarily adapted for use in knot
State is configured with the simple bridge type of stress fixation into the bridge stage, can not really be reflected in complicated bridge type construction dynamic process
Structural form and the changing rule that is influenced by Shear Lag of stress.
105046027 A of Chinese patent literature CN discloses a kind of more ribbed tee girder bridge sections on November 11st, 2015
Optimum design method, it establishes the control differential equation and nature of more ribbed tee girder bridges based on minimum potential energy principal
Boundary condition considers that more ribbed beam bridge Shear Lag Effects, shear lag warping stress self-balancing condition and hophornbeam Xin Ke cut shear
The influence of the factors such as shape to the mechanical analysis that the class formation is refined, and then is made by the selection of rational cross section size more
Ribbed beam bridge is in good mechanical state.105243236 A of Chinese patent literature CN discloses one on January 13rd, 2016
Kind box-girder bridge section design optimization method, this method consider the shadow of the factors such as box-girder shear lag and shear-deformable effect
It rings, acquires the potential energy of deformation and kinetic energy of box-girder first, and then obtain structural dynamic control differential side using Hamiton's principle
Journey and natural boundary conditions.But the shortcomings that the two patents, is：They are confined to Constructional differential equation and side derived from energy method
Boundary's condition still can not be applied to the analysis of Shear Lag of complicated Construction of Cable-Stayed Bridges.
" bending the finite segment method that box beam Shear Lag calculates ", sieve flag, Foshan Univ.'s journal, the 2nd phase of volume 12, the
Page 23~31, one kind are described in April, 1994 under Compact-bending Load collective effect, using the homogeneous solution of control differential equation as cutting
The stagnant warp displacement mode of power, the finite segment method calculated suitable for cable-stayed bridge Shear Lag." the one of box-girder Analysis of Shear Lag Effect
Tie up FInite Element and its application ", Zhang Yuanhai, Wang Lailin, Li Qiao, civil engineering journal, the 8th phase of volume 43, page 44~50,
In August, 2010 describes a kind of improved the finite segment method, can by introducing shear lag Generalized Moment and warpage geometrical property
Analyze the Girder with Shear Lag Effect of the labyrinths such as box girder with variable cross section and special support continuous box girder.But these methods are equal on each node
Only consider that a Shear Lag freedom degree, Shear Lag freedom degree refer to the independent node position introduced for describing Shear Lag state
It moves, sieve, two people only consider a Shear Lag freedom degreeIt cannot be adapted to the boundary condition of various analysis of Shear Lag.
Bondbeam, which is girder steel and concrete slab, combines the common stress in section formed, compatibility of deformation by shear connector
A kind of composite structure, also known as combination beam (highway Railway System habit uses " bondbeam ", other professions claim " combination beam " more).Such as figure
Shown in 1, Π shape bondbeam refers to the combination beam that concrete slab 1 and girder steel 2 are combined on cross section with the shape of alphabetical Π.
For П shape combination girder stayed-cable bridge, (suspension cable anchor point bias and in advance answered since girder bears concentrated bending moment
Power anchorage is eccentric) and the Shear Lag displacement boundary conditions mutation problems that generate, load and Shear Lag displacement in dynamic process of constructing
The continually changing problem of boundary condition, the above method can not be fully solved.Since the prior art is confined under single operating condition
Simple structure analysis, not can solve in CONSTRUCTION OF CABLE-STAYED BRIDGE dynamic process because there are the construction errors caused by Shear Lag Effect to ask
Topic.
Summary of the invention
For technical problem of the existing technology, the present invention provides a kind of processing sides of Π shape bondbeam Shear Lag
Method, it can consider in real time influence of the Shear Lag to girder stress and deformation in simulation CONSTRUCTION OF CABLE-STAYED BRIDGE dynamic process, improve
The accuracy of Π shape bondbeam Construction of Cable-Stayed Bridges.
It is realized the technical problem to be solved by the present invention is to technical solution in this way, it includes
Step 1, according to the design feature of Π shape composite beam section, construction is suitable for the Shear Lag warpage position of Π shape bondbeam
Move function ui(x, y) is：
In formula,For Shear Lag displacement；
hiFor the distance of concrete slab or the middle plane of steel beam flange to Π shape composite beam section neutral axis；
ζiFor the correction factor for considering shear-deformable influence degree, takes 1-to consider, takes 0-not consider；
B is the 1/2 of two steel beam web plate spacing；
Subscript i=1,2,3,4 indicate respectively tetra- regions 1#, 2#, 3#, 4# of Π shape composite beam section：
The region 1# is the half of clear distance range inner concrete plate between two girder steels；
The region 2# be girder steel top flange top concrete plate and overhanging unilateral concrete slab and；
The region 3# is girder steel lower flange；
The region 4# is girder steel top flange；
ξiFor the width correction factor, the respectively ratio of the width and b of 1#~4# zone plate；
Step 2, according to the Shear Lag warp displacement function u established in step 1i(x, y) calculates total gesture of Π shape bondbeam
It can Π；
Step 3, it according to the total potential energy Π acquired in step 2, is expressed by minimum potential energy principal foundation with variational form quiet
Equilibrium equation, the equation of static equilibrium are using displacement function as fundamental unknown variables；
Step 4, the equation of static equilibrium is solved, each node introduces the positional displacement interpolation function of two Shear Lag freedom degrees, right
The resulting equation of static equilibrium of step 3 carries out discretization, is deformed into using generalized nodal displacement as the balance side of fundamental unknown variables
Journey obtains the element stiffness matrix [K] for considering that Shear Lag influences.
Particularly, in step 2, the total potential energy of Π shape bondbeam is：
In above formula,For Shear Lag displacementBelong to angular displacement；For Shear Lag displacementFirst derivative, belong to
Generalized displacement.
Technical effect of the invention：
Since the present invention proposes reasonable Shear Lag warp displacement mode by the cross-section analysis to Π shape bondbeam, and
Intercoupling between vertical displacement and Shear Lag displacement is considered, uses two Shear Lag freedom degrees at each section of Π shape bondbeam
(i.e.With), it can adapt to different Shear Lag displacement boundary conditions.In addition, all modeling process all programs
Change, convenient for operation and programming, greatly reduced artificial calculation amount, the meter of Π shape bondbeam Shear Lag is realized by computer
It calculates, the changing rule of Π shape girder Shear Lag and influence in complicated CONSTRUCTION OF CABLE-STAYED BRIDGE dynamic process is studied with this.
So it is an advantage of the invention that：Different Shear Lag displacement boundaries is adapted to, Π shape combination girder stayed-cable bridge is improved
The accuracy of work progress more meets reality to the prediction result of girder stress and deformation.
Detailed description of the invention
Detailed description of the invention of the invention is as follows：
Fig. 1 is the structural schematic diagram of Π shape composite beam section；
In Fig. 1：1. concrete slab；2. girder steel；
Fig. 2 is the direct stress distributional assumption figure of one unit of Π shape bondbeam of the present invention；
Fig. 3 is the overall structure diagram of cable-stayed bridge main-beam；
The structural parameters that Fig. 4 is Fig. 1 mark figure；
Fig. 5 is the Stayed Cable Bridge of one embodiment across layout drawing；
Fig. 6 is that the cable-stayed bridge cross sectional dimensions of one embodiment marks figure；
Fig. 7 is the stagnant coefficient of The East Pagoda end bay auxiliary pier pier top section shear in embodiment with Construction State variation diagram；
Fig. 8 is the schematic diagram of CONSTRUCTION OF CABLE-STAYED BRIDGE sequence and section number, beam element structure；
Fig. 9 is that the actual measurement of The East Pagoda end bay auxiliary pier pier top section, theoretical stress compare after the installation of 10# floorings in embodiment
Figure；
Figure 10 is the absolute altitude difference curve graph of girder after 14# cable tension in embodiment.
Specific embodiment
Present invention assumes that the direct stress of one unit of Π shape bondbeam is distributed as shown in Fig. 2, establishing unit coordinate in Fig. 2
System's (or being local coordinate system), it is vertically the direction z that axial (length direction), which is the direction x, and laterally (width direction) is the direction y.
The maximum difference of shear rotation angle is shown in Fig. 2
As shown in figure 3, cable-stayed bridge main-beam is that unit one as shown in Figure 2 spells successively, it, can for overall structure
Establish a global coordinate system (or being structure coordinate system).In global coordinate system, along bridge to (also known as vertical bridge is to length side
To) it is X-direction, direction across bridge (width direction) is Y-direction, is vertically Z-direction to (short transverse).
The present invention be directed to a kind of beam elements, and mainly the formation of unit is derived and described, therefore need to be to unit coordinate
System is arranged.Structure coordinate system can be different with the difference of different software and reckoner, only relate to the conversion of finite element coordinate
Problem, not within the scope of creation of the invention.
Present invention will be further explained below with reference to the attached drawings and examples, and the present invention includes the following steps：
Step 1：The characteristics of being constructed according to Π shape composite beam section --- concrete slab width is much larger than girder steel between two girder steels
With the width on the concrete edge of a wing on girder steel, during Π shape combination beam deformed, concrete slab is due to shear-deformable
Caused by shear lag phenomenon it is more significant for the concrete edge of a wing；Simultaneously as concrete material is a kind of non-
Uniformly, anisotropic material with complex discusses merely that the stress of certain point is not anticipate from micro-scale in the actual process
It is adopted and unnecessary, and the mean stress of the macro-regions big relative to aggregate size several times is discussed and just more meets reality.It is based on
These characteristics, for the Π shape bondbeam that girder steel and concrete slab are well combined, bond area is smaller, region internal stress level
Be not much different, it is believed that the stagnant influence very little of the region internal shear force and ignore its influence, so, this step proposition be suitable for Π shape knot
Close the Shear Lag warp displacement function u of beami(x,y)：Only consider girder steel between clear distance range inner concrete plate shear lageffect, suddenly
The slightly influence of remaining each section Shear Lag.
Notation convention, and selected Shear Lag warp displacement mould are carried out to the structure composition of Π shape bondbeam cross section first
Formula.For convenience of description, Π shape composite beam section is divided into 4 regions, as shown in Figure 1：
The region 1# --- the half of clear distance range inner concrete plate between two girder steels, subscript are indicated with " 1 "；
The region 2# --- girder steel top flange top concrete plate and overhanging unilateral concrete slab and, subscript use
" 2 " indicate；
The region 3# --- girder steel lower flange, subscript are indicated with " 3 "；
The region 4# --- girder steel top flange, subscript are indicated with " 4 "；
For steel beam web plate, the subscript of dependent cross-section geometrical property is indicated with " w ".
As shown in figure 4, taking the 1/2 of two steel beam web plate spacing to be used as width b, then ξiB (i=1,2,3,4) is the area 1#~4#
Domain plate width, wherein ξiThe respectively ratio of the width and b of 1#~4# zone plate, referred to as the width correction factor.In Fig. 4, t1、
t2、t3、t4The respectively thickness of 1#~4# zone plate plate, h1、h2Plane is to Π shape knot respectively in the concrete slab of the region 1#, 2#
Close the distance of beam section neutral axis；h3、h4For the distance of plane in the upper and lower edge of a wing of girder steel to Π shape composite beam section neutral axis.
It introduces axial displacement u=u (x), vertical displacement v=v (x) and Shear Lag warp displacement uiThe three broad sense positions (x, y)
It moves.Direct stress distribution as shown in Figure 2, ignores the shear lageffect in the region 2#~4#, only considers the shear lageffect in the region 1#,
It is, by u1(x, y) is assumed to 3 parabolic distributions, and u2(x,y)、u3(x,y)、u4(x, y) is taken as horizontal linear, i.e.,
In formula (1),For the maximum difference of shear rotation angle, also known as Shear Lag is displaced；
hiFor the distance of concrete slab or the middle plane of steel beam flange to Π shape composite beam section neutral axis；
ζiThe correction factor for considering shear-deformable influence degree, only has herein：1-considers that 0-does not consider.
Formula (1) be Π shape bondbeam Shear Lag warp displacement mode, it should be noted that dv/dx withIt is contrary sign, in 1#
Region and the region 2# intersection displacement components ui(x, y)=hiDv/dx is maximum value (region 1# and the region 2# intersection u1(x, y)=
u2(x,y))。
Web meets the plane cross-section assumption of General beam theory under symmetric curvature load action.It only relates to herein longitudinal curved
Bent deformation potential and axial deformation deformation potential, and transverse curvature deformation potential is negligible, the vertical extruding of upper lower flange
εz, the out-of-plane shear-deformable γ in the edge of a wingxz, γyzAnd transverse curvature, transverse strain belong to micro, can be neglected.
Step 2：According to the Shear Lag warp displacement function u established in step 1i(x, y) calculates total potential energy of structural system
Π includes：Load potential energy V, girder axial deformation deformation potential UA, girder Bending Deformation potential energy (Uw+Uc+Us), wherein UwFor steel
Web Bending Deformation potential energy, UcFor concrete slab Bending Deformation potential energy, UsFor steel beam flange Bending Deformation potential energy.The axis
Refer to that displacement of the neutral plane along unit axis, the bending deformation refer to corner of the cross section around neutral axis to deformation.
Various potential energy are calculated separately below：
(1) when beam is by " unit evenly load " and " node load ", load potential energy V is
In formula (2), { Δ } is modal displacement array, and { F } is nodal load matrix, and q (x) is unit vertical uniform load, l
For element length.
(2) girder axial deformation deformation potential UA
In formula (3), EcFor modulus of elasticity of concrete, EsFor the elasticity modulus of steel；U '=u ' (x) is axial displacement u=u (x)
First derivative, AcFor concrete slab area of section, (section is the cross section of beam element, for one dimension finite element, is pressed
Agreement geometric characteristics refer in particular to the geometric characteristics in cell cross-section), AsFor girder steel area of section, A=E is enabledcAc/
Es+As, have：
(3) girder Bending Deformation potential energy
Steel beam web plate
In formula (5), IwIt is web to the moment of inertia of Π shape composite beam section neutral axis, v " is the second order of vertical displacement v=v (x)
Derivative.
Concrete slab
In formula (6), GcFor the modulus of shearing of concrete, U1For the Bending Deformation potential energy of the region 1# concrete slab, U2For the area 2#
The Bending Deformation potential energy of domain concrete slab.
Steel beam flange
In formula (7), GsFor the modulus of shearing of steel, U3For the Bending Deformation potential energy of the region 3# girder steel lower flange, U4For the region 4#
The Bending Deformation potential energy of girder steel top flange.
In formula (6) and formula (7), tiFor the thickness of the region i# plate, εi、γiRespectively
Formula (8) is substituted into formula (6), formula (7) respectively, is had
1. concrete slab Bending Deformation potential energy Uc
It enablesSimultaneously
Ica=I1+I2,Icb=ζ1I1,
It can obtain
In formula (12),Referred to as " generalized nodal displacement " can describe independent degree on cell node in a broad sense
Variable, being corresponding to it has " generalized nodal force ".It is about the reasons why " generalized nodal displacement " formation：
Any point in two-dimensional surface should only there are two independent modal displacement u, v (the two displacements are displacements of the lines),
But for member structure, corner displacement θ (i.e. angular displacement) is introduced on node, to describe the bending change of rod piece, so
There are 3 displacement components us, v, θ, and θ=dv/dx=v on each node of general plane bar element/, this is that civil engineering basis is known
Know.
In order to describe the shear lageffect of thin-walled bar, the present invention above-mentioned bar unit on the basis of introducing two changes again
Measure φ andTo describe shear lageffect, wherein φ is the maximum difference of shear rotation angle, identical as the dimension of rotational angle theta, is belonged to logical
Normal " angular displacement " scope；And φ isFirst derivative, do not correspond to a certain specific change in displacement of bar element, belong to wide
The scope that adopted position is moved.Mathematically, it is the displacement state in description bar unit vertical plane, can theoretically following variables is taken to make
For modal displacement：
V, θ=v/、v//、v///、……、v(n)
But only the first two displacement is for specific change in displacement, v --- displacement of the lines, θ=v/--- angular displacement, and thereafter
V//、v///、……、v(n)It is only mathematically significant, and be corresponding to it without actual displacement, therefore referred to as generalized displacement.
2. steel beam flange Bending Deformation potential energy Us
It enablesSimultaneously
Isa=2I3+2I4,Isb=Isc=Isd=0 (15)
It can obtain
It enables
It can obtain
According to formula (4), formula (5) and formula (18) it is found that Π shape bondbeam deformation potential U is
In formula (19), I=Iw+Ifa。
By formula (2) and formula (19) it is found that the total potential energy Π of Π shape bondbeam is
In formula (20),
V is load potential energy；
U is Π shape bondbeam deformation potential (including girder axial deformation deformation potential UA, steel beam web plate Bending Deformation potential energy
Uw, concrete slab Bending Deformation potential energy UcWith steel beam flange Bending Deformation potential energy Us)；
A=EcAc/Es+As；
EsFor the elasticity modulus of steel；
EcFor modulus of elasticity of concrete；
AcFor concrete slab area of section；
AsFor girder steel area of section；
L is element length；
U '=u ' (x) is the first derivative of axial displacement u=u (x)；
V (x) is vertical displacement, and ν " is the second dervative of vertical displacement v=v (x)；
For Shear Lag displacement For Shear Lag displacementFirst derivative；
{ Δ } is modal displacement array, and { F } is nodal load matrix, and q (x) is unit vertical uniform load；
GsFor the modulus of shearing of steel；
B is the 1/2 of two steel beam web plate spacing；
I=Iw+Ifa
IwIt is web to the moment of inertia of Π shape composite beam section neutral axis；
Ica=I1+I2, Icb=ζ1I1,
Isa=2I3+2I4, Isb=Isc=Isd=0
t1、t2、t3、t4The respectively thickness of 1#~4# zone plate plate；
h1、h2Respectively in the concrete slab of the region 1#, 2# plane to Π shape composite beam section neutral axis distance；
h3、h4Respectively in the steel beam flange of the region 3#, 4# plane to Π shape composite beam section neutral axis distance；
ξiB (i=1,2,3,4) is the region 1#~4# plate width.
Step 3：According to minimum potential energy principal, under external force, the elastomer in stability is meeting side
In all displacements of boundary's condition, unique one group of displacement there are in fact, this group displacement can make total potential energy of whole system most
Small, i.e. the first variation of the total potential energy of system should be zero
δ Π=δ (U+V)=0 (21)
Formula (20) are substituted into, can be obtained
Formula (22) is the total potential energy Π acquired in foundation step 2, and is expressed by minimum potential energy principal foundation with variational form
The equation of static equilibrium, the equation be with u (x), v (x) andThree displacement functions are as fundamental unknown variables.
It should be noted that the displacement function of selection must meet displacement boundary conditions on being displaced known boundary,
Face force boundary is then not necessarily to consider, because of its automatic satisfaction.
Step 4：In meeting cable-stayed bridge under Shear Lag displacement boundary conditions sudden change conditions caused by concentrated bending moment, step is solved
Rapid 3 resulting equilibrium equation.
Beam element structure as shown in Figure 8：According to the requirement of beam section finite element, need to serve as reasons overall structure discretization " section
The finite element model of point+beam element " composition, also known as finite element grid, each beam element have 2 nodes --- the end i and the end j.This
In step, each node introduces two Shear Lag freedom degrees (i.e.With) positional displacement interpolation function.
It is introduced into positional displacement interpolation function and modal displacement (10 displacement parameters i.e. in modal displacement array { Δ }), to formula
(22) discretization is carried out.Axial displacement u uses linear interpolation, vertical displacement v and Shear Lag displacement3 Hermite is selected to insert
Value, shape function use [N respectivelyu]、[Nv] andIt indicates
U=[Nu] { Δ }, v=[Nv]{Δ},
In formula (23)
Modal displacement array { Δ } includes 10 displacement parameters, is
Nodal load matrix corresponding with formula (26) is
{ F }=[Ni Qi Mi Si Ti Nj Qj Mj Sj Tj]T (27)
In formula (26) and formula (27), i, j are the end i and the end j of beam element；uiAnd ujFor axial displacement；viAnd vjFor vertical position
It moves；θiAnd θjFor angular displacement；WithFor the generalized shear stagnating bit shifting parameter at unit both ends；NiAnd NjFor axle power；QiWith
QjFor shearing；MiAnd MjFor moment of flexure；Si、Ti、SjAnd TjRespectively withWithThe corresponding stagnant unit node of generalized shear
Power.
It brings formula (23) into formula (22), has
Because node virtual displacement { δ Δ } is arbitrary value, have
It enables
Then formula (29) is rewritten as the equilibrium equation using generalized nodal displacement as fundamental unknown variables：
In formula (31), [K]=[Ke]+[Ks] it is the element stiffness matrix for considering Shear Lag and influencing；{ Δ } is modal displacement column
Battle array；{ F } is node load array；For unit evenly load array.
[Ke] be existing elementary beam element elastic stiffness matrix
Shear Lag and curved coupling influence are by [Ks] indicate
Formula (24), formula (25) are substituted into formula (30), obtain considering the element stiffness square that Shear Lag influences under unit coordinate system
Battle array [K] be：
Finally, being coordinately transformed and always rigid group using element stiffness matrix of the existing finite element method to formula (34)
Collection, to solve the equilibrium equation based on modal displacement.
Formula (31) is unit analysis as a result, element stiffness matrix therein [K] (abbreviation Dan Gang) is a singular matrix, nothing
Method solves.To acquire modal displacement, global analysis is carried out, needs to be coordinately transformed to obtain under global coordinate system by formula (31)
Dan Gang [K], and a group collection is carried out to each unit stiffness matrix and obtains the global stiffness matrix of bed rearrangement cable-stayed bridge, could formed whole
The nodal equilibrium equation of body structure, the equation are exactly the equilibrium equation based on modal displacement that can be solved, and then it is total to form structure
Body stiffness matrix (referred to as total rigid) and its equilibrium equation, this belongs to the generalized flowsheet of finite element method, is obtained using existing algorithm
As a result.
Embodiment
Fig. 5 is a Stayed Cable Bridge across layout drawing, and Fig. 6 is cable-stayed bridge cross sectional dimensions mark figure.It is provided by Fig. 5 and Fig. 6
Cable-stayed bridge span distribution and section parameter are shown in Table 1, wherein each beam section steel plate plate dimensions is shown in Table 2.
1 Stayed Cable Bridge of table is across size and section parameter
Each beam section steel plate board dimensions (mm) of table 2
Explanation：2 central sill segment type of table is numbered for being identified to different types of segment.One CONSTRUCTION OF CABLE-STAYED BRIDGE circulation
One segment of middle formation, and all parts size of each segment can change with demand, in order to each type of segment
It is identified, introduces beam section type number and the beam section with identical size is numbered.
Each beam section cross section geometric characteristic parameter is calculated according to formula (17)：A, I, Ifb, IfcAnd Ifd, the results are shown in tables 3.It is mixed
Solidifying soil elasticity modulus is Ec=35500MPa, steel elasticity modulus are Es=210000MPa, the half of steel beam web plate spacing are b=
12m。
Each beam section cross section geometric characteristic parameter (mm of table 32,mm4)
By the cross section geometric characteristic parameter in table 3, modulus of elasticity of concrete Ec, steel elastic modulus EsAnd steel beam web plate
Half b and unit computational length l of spacing substitute into the element stiffness matrix [K] for considering that Shear Lag influences, and can acquire each beam section and consider
The element stiffness matrix that Shear Lag influences.
Show applying in cable-stayed bridge by the sunykatuib analysis to practical Π shape combination girder stayed-cable bridge project construction dynamic process
During work, the stress behavior of Π shape bondbeam shows apparent shear lag phenomenon.It should be noted that arbitrary section
For Shear Lag it is not unalterable, but as the propulsion of work progress and the differentiation of structural form constantly change, such as Fig. 7
It is shown.Abscissa indicates that construction procedure number, ordinate are shear lag coefficient λ in Fig. 7, and the calculating of shear lag coefficient λ is as follows：
In formula (35), σx、Indicate cross section stresses,For elementary beam element section stress, σxConsider for the present invention
Section stress after shear lageffect.
In formula (36), M (x) is section turn moment.In the region 1# and the region 2# intersection (y=ξ1B) shear lag coefficient is
It is in concrete slab midpoint (y=0) shear lag coefficient
It will consider that element stiffness matrix [K] the substitution general finite element program that Shear Lag influences calculates, acquire broad sense
Modal displacementAnd formula (38), (39) are substituted into, λ is acquired to calculatewAnd λc。
As seen from Figure 7, in the construction process, the stagnant change for not only numerically having size of same section shear, in mode
It is upper that there are also the Alternate Phenomenons of positive and negative Shear Lag.
As shown in figure 8, Π shape combination girder stayed-cable bridge is assembled, the bridge surface in bridge project of the segment of segment one
The segment Ta Chu is about set to 0#, then with the formation of girder, newly-increased segment successively according to 1#, 2#, 3# ..., i# compiled
Number, this is the general agreement in science of bridge building.
For the construction effect for verifying present method invention, for The East Pagoda end bay auxiliary pier pier top section as shown in Figure 5, applying
Work stage 10# floorings installation after, with the result of General beam theory calculated result, calculated result of the invention and actual test by
Shown in Fig. 9, ordinate is section stress in Fig. 9, and abscissa is direction across bridge coordinate, is found out by Fig. 9：Present method invention is calculated
Section theoretical stress closer to actual measured results, it is higher than the accuracy that application General beam theory calculates.
The Stress calculation of General beam theory is according to obtained by formula (36), and Stress calculation of the invention is according to obtained by formula (37).By
The coupling influence of Shear Lag Yu girder bending stiffness has been fully considered in the present invention, so can inquire into Π shape bondbeam oblique pull
Influence of the Shear Lag to girder bending stiffness during bridge construction, to realize linear accurate of Construction of Cable-Stayed Bridges middle girder
Control.
Figure 10 is the absolute altitude difference curve graph of girder after 14# cable tension.14# drag-line is the longest suspension cable in east, Xi Ta,
Absolute altitude difference is to calculate height value (specially by the present invention：It will consider that element stiffness matrix [K] substitution that Shear Lag influences is general
Finite element program is calculated, and acquires node theory vertical displacement v), then subtracted each other with the absolute altitude value of actual measurement and found out.
Found out by Figure 10：Maximum cantilever state after 14# cable tension, the theoretical absolute altitude calculated using present method invention
And the difference very little of measured value is no more than 3cm, and result is positive and negative staggeredly shows certain randomness and unsystematic defect, table
The bright present invention accurately reflects the case where Π shape bondbeam actual loading deformation.
Claims (5)
1. a kind of processing method of Π shape bondbeam Shear Lag, characterized in that include the following steps：
Step 1, according to the design feature of Π shape composite beam section, construction is suitable for the Shear Lag warp displacement letter of Π shape bondbeam
Number ui(x, y) is：
In formula,For Shear Lag displacement；
hiFor the distance of concrete slab or the middle plane of steel beam flange to Π shape composite beam section neutral axis；
ζiFor the correction factor for considering shear-deformable influence degree, takes 1-to consider, takes 0-not consider；
B is the 1/2 of two steel beam web plate spacing；
Subscript i=1,2,3,4 indicate respectively tetra- regions 1#, 2#, 3#, 4# of Π shape composite beam section：
The region 1# is the half of clear distance range inner concrete plate between two girder steels；
The region 2# be girder steel top flange top concrete plate and overhanging unilateral concrete slab and；
The region 3# is girder steel lower flange；
The region 4# is girder steel top flange；
ξiFor the width correction factor, the respectively ratio of the width of the region 1#~4# plate and b；
Step 2, according to the Shear Lag warp displacement function u established in step 1i(x, y) calculates total potential energy Π of structural system；
Step 3, it according to the total potential energy Π acquired in step 2, is put down by minimum potential energy principal foundation with the static(al) that variational form is expressed
Weigh equation, and the equation of static equilibrium is using displacement function as fundamental unknown variables；
Step 4, the equation of static equilibrium is solved, each node introduces the positional displacement interpolation function of two Shear Lag freedom degrees, to step 3
The resulting equation of static equilibrium carries out discretization, is deformed into the equilibrium equation using generalized nodal displacement as fundamental unknown variables, obtains
It must consider the element stiffness matrix [K] that Shear Lag influences.
2. the processing method of Π shape bondbeam Shear Lag according to claim 1, it is characterized in that：In step 2, the Π
The total potential energy of shape bondbeam is：
In formula,
V is load potential energy；
U is Π shape bondbeam deformation potential；
A=EcAc/Es+As；
EsFor the elasticity modulus of steel；
EcFor modulus of elasticity of concrete；
AcFor concrete slab area of section；
AsFor girder steel area of section；
L is element length；
U '=u ' (x) is the first derivative of axial displacement u=u (x)；
V (x) is vertical displacement, and ν " is the second dervative of vertical displacement v=v (x)；
For Shear Lag displacement For Shear Lag displacementFirst derivative；
{ △ } is modal displacement array, and { F } is nodal load matrix, and q (x) is unit vertical uniform load；
GsFor the modulus of shearing of steel；
B is the 1/2 of two steel beam web plate spacing；
I=Iw+Ifa
IwIt is web to the moment of inertia of Π shape composite beam section neutral axis；
Ica=I1+I2, Icb=ζ1I1,
Isa=2I3+2I4, Isb=Isc=Isd=0
t1、t2、t3、t4The respectively thickness of the region 1#~4# plate；
h1、h2Respectively in the concrete slab of the region 1#, 2# plane to Π shape composite beam section neutral axis distance；
h3、h4Respectively in the steel beam flange of the region 3#, 4# plane to Π shape composite beam section neutral axis distance；
ξ1b、ξ2b、ξ3b、ξ4B is the width of the region 1#~4# plate.
3. the processing method of Π shape bondbeam Shear Lag according to claim 2, it is characterized in that：In step 3, described quiet
Equilibrium equation is：
Equation be with u (x), v (x) andThree displacement functions are as fundamental unknown variables.
4. the processing method of Π shape bondbeam Shear Lag according to claim 3, it is characterized in that：In step 4, it is described with
Generalized nodal displacement is as the equilibrium equation of fundamental unknown variables：
In formula,For unit evenly load array；
[K]=[Ke]+[Ks] it is the element stiffness matrix for considering Shear Lag and influencing；
[Ke] be existing elementary beam element elastic stiffness matrix；
[Ks] be Shear Lag and curved coupling influence stiffness matrix；
5. the processing method of Π shape bondbeam Shear Lag according to claim 4, it is characterized in that：In step 4, described to examine
Consider Shear Lag influence element stiffness matrix [K] be：
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| CN115345047A (en) * | 2022-08-11 | 2022-11-15 | 南京理工大学 | Space beam unit construction method considering shear hysteresis effect of wide box beam |
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